I have a 20-year old 26" rim with a new Schwalbe Big Apple tire on it. When I first mounted the tire and put them on a bike, the front wheel was wobbly. I assumed this was because the 20 year old rim was out of true. I basically shelved the bike and only took it out for aound the block spins to try new things out (ie, handlebars, stems, etc..). I finally got around to taking the wheel/tire to an LBS for truing and the mechanic came back and said the wheel was true and the wobble was in the tire. He reinstalled the tire and it made a considerable difference but the tire still wobbles a little bit. His advice was to "just ride it". Easier said than done because riding a bike with a wobbly front end just isn't fun.

I went back to Amazon.com to return the tire and the time limit has expired so I'm stuck with the tire. Is there anything that can be done to correct this problem? I don't even know what the particular defect is but my limited understanding leads me to suspect the bead.

There is typically a thin mold line on the tire about 5mm or so from the rim bead.
Examine that to see that it is concentric with the rim.
IF not, you need to take more care in seating the tire.
If it looks nice & concentric, you have a bad tire.

Without more info, it's impossible to know whether it's a mounting and seating issue, or a tire defect.

Spin the wheel slowly and keep an eye on the reference line molded into the tire just above the rim. It should be equidistant from the rim all the way around. Check both sides, and if the like moves up and down, deflate the tire down to about 5-10psi, and massage it around until the reference lines run true on both sides, then reinflate.

OTOH- if the reference line is true, but the tire wobbles you're SOL and the solution is a new tire.

BTW- if you end up replacing the tire, be aware that the wider the tire is compared to the rim (ie. beyond 1.5x the rim width) the more sensitive it is to seating, and prone to the wobble problem.

I'd prefer to put any questionable tire on the back wheel since a sudden failure there is less likely to cause a crash. And in this case I'd expect the slight wobble to be less noticeable on the rear wheel than on the front where it impacts your steering. If you can't return it I'd move it to the rear rim and see if you can get some use out of it that way.

How much wobble are we talking here? Does the bike actually wobble, or is it just a tiny visual thing? I ask because I have never had a perfect 26 inch bike tire in forty years and a zillion brands. Yes, a zillion. But seriously, every one has had a mm this way or 2 mm that way. I have always checked the reference line and it has always been kosher.

Couple of things, patched tubes may have some lumps. Check to make sure rim strip is level and even. Then put on the tube and tire. Take a soapy sponge and apply to both beads, inflate using an air compressor, usually this helps seating new tires evenly.

I checked the reference line and it was off a bit. I had assumed that when the LBS guy put in a new tube, he would have eliminated this variable. I deflated the tire, used my fingers stuff the bead all around the perimeter on both sides of the tire and aired it back up. It is still not perfect but it is a significant improvement. The wobble is not just visible, it's felt in the front-end of the bike shaking. At this point it is so subtle that I have to pay attention to feel it. Of course, the wind is blowing at about a steady 20+ mph right now and that screws everything up with my perception of how a bike is operating. I'll get it out on a windless day and see how it feels. I'm now optimistic where before I was ready to write off this tire as useable.